Bump Maps vs Normal Maps
Bump maps and normal maps are essentially the same thing.
The primary difference is that normal maps have more information in them(because they use an rgb input) to give a more accurate bump effect.
The rgb information in the normal maps correspond to the x,y,z axis. 

Bump maps use only a black to white map to understand the depth.

Both bumps and normals are essentially a way to cheat the shading to give the effect of depth...meaning no resolution is added to the geometry in anyway... to do that that we use displacement maps.
Displacement maps unlike the other two actually modify the geometry at render time.